{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "4LqvmtZPzyY1"
      },
      "source": [
        "[![Roboflow Notebooks](https://media.roboflow.com/notebooks/template/bannertest2-2.png?ik-sdk-version=javascript-1.4.3&updatedAt=1672932710194)](https://github.com/roboflow/notebooks)\n",
        "\n",
        "# Fine-tune PaliGemma2 on LaTeX OCR Dataset\n",
        "\n",
        "---\n",
        "\n",
        "[![GitHub](https://badges.aleen42.com/src/github.svg)](https://github.com/google-research/big_vision/blob/main/big_vision/configs/proj/paligemma/README.md)\n",
        "[![arXiv](https://img.shields.io/badge/arXiv-2412.03555-b31b1b.svg)](https://arxiv.org/abs/2412.03555)\n",
        "[![Roboflow](https://raw.githubusercontent.com/roboflow-ai/notebooks/main/assets/badges/roboflow-blogpost.svg)](https://blog.roboflow.com/fine-tune-paligemma-2/)\n",
        "\n",
        "PaliGemma 2 is built by combining the SigLIP-So400m vision encoder with the more recent and capable language models from the Gemma 2 family.\n",
        "\n",
        "![PaliGemma2 Figure.1](https://storage.googleapis.com/com-roboflow-marketing/notebooks/examples/paligemma2-1.png)\n",
        "\n",
        "The authors use a 3-stage training approach similar to the original PaliGemma. In stage 1, they combine the pretrained vision and language model components and train them jointly on a multimodal task mixture. In stage 2, they train the models at higher resolutions of 448px^2 and 896px^2. In stage 3, they fine-tune the models on the target transfer tasks.\n",
        "\n",
        "PaliGemma 2 models outperform the original PaliGemma at the same resolution and model size. Increasing the model size and resolution generally improves performance across a wide range of tasks, but the benefits differ depending on the task. Some tasks benefit more from increased resolution, while others benefit more from a larger language model.\n",
        "\n",
        "![PaliGemma2 Figure.2](https://storage.googleapis.com/com-roboflow-marketing/notebooks/examples/paligemma2-2.png)\n",
        "\n",
        "Notebook requires A100 with 40GB of VRAM to train."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "lBp3Czz3GBmc"
      },
      "source": [
        "## Setup"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ADTkh-2y_9Yv"
      },
      "source": [
        "### Configure your API keys\n",
        "\n",
        "To fine-tune PaliGemma2, you need to provide your HuggingFace Token and Roboflow API key. Follow these steps:\n",
        "\n",
        "- Open your [`HuggingFace Settings`](https://huggingface.co/settings) page. Click `Access Tokens` then `New Token` to generate new token.\n",
        "- Go to your [`Roboflow Settings`](https://app.roboflow.com/settings/api) page. Click `Copy`. This will place your private key in the clipboard.\n",
        "- In Colab, go to the left pane and click on `Secrets` (🔑).\n",
        "    - Store HuggingFace Access Token under the name `HF_TOKEN`.\n",
        "    - Store Roboflow API Key under the name `ROBOFLOW_API_KEY`."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "4wyojKiG_hX9"
      },
      "source": [
        "### Select the runtime\n",
        "\n",
        "Let's make sure that we have access to GPU. We can use `nvidia-smi` command to do that. In case of any problems navigate to `Edit` -> `Notebook settings` -> `Hardware accelerator`, set it to `A100 GPU`, and then click `Save`."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "O_8BLW6R_x-z",
        "outputId": "785d09a8-5ab1-4a59-ed6b-7fdba1f52420"
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Wed Dec 11 13:02:13 2024       \n",
            "+---------------------------------------------------------------------------------------+\n",
            "| NVIDIA-SMI 535.104.05             Driver Version: 535.104.05   CUDA Version: 12.2     |\n",
            "|-----------------------------------------+----------------------+----------------------+\n",
            "| GPU  Name                 Persistence-M | Bus-Id        Disp.A | Volatile Uncorr. ECC |\n",
            "| Fan  Temp   Perf          Pwr:Usage/Cap |         Memory-Usage | GPU-Util  Compute M. |\n",
            "|                                         |                      |               MIG M. |\n",
            "|=========================================+======================+======================|\n",
            "|   0  NVIDIA A100-SXM4-40GB          Off | 00000000:00:04.0 Off |                    0 |\n",
            "| N/A   31C    P0              43W / 400W |      2MiB / 40960MiB |      0%      Default |\n",
            "|                                         |                      |             Disabled |\n",
            "+-----------------------------------------+----------------------+----------------------+\n",
            "                                                                                         \n",
            "+---------------------------------------------------------------------------------------+\n",
            "| Processes:                                                                            |\n",
            "|  GPU   GI   CI        PID   Type   Process name                            GPU Memory |\n",
            "|        ID   ID                                                             Usage      |\n",
            "|=======================================================================================|\n",
            "|  No running processes found                                                           |\n",
            "+---------------------------------------------------------------------------------------+\n"
          ]
        }
      ],
      "source": [
        "!nvidia-smi"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "FMlw3ru1YvLg"
      },
      "source": [
        "### Download dataset from Roboflow Universe\n",
        "\n",
        "To fine-tune PaliGemma2, prepare your dataset in JSONL format. You can use Roboflow to easily convert any dataset into this format."
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "!pip install -q roboflow supervision peft bitsandbytes"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "2A-B4zT3_T3Q",
        "outputId": "f9e9b00f-e34e-429f-bfbf-a64839300874"
      },
      "execution_count": 2,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "\u001b[?25l   \u001b[90m━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━\u001b[0m \u001b[32m0.0/80.9 kB\u001b[0m \u001b[31m?\u001b[0m eta \u001b[36m-:--:--\u001b[0m\r\u001b[2K   \u001b[90m━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━\u001b[0m \u001b[32m80.9/80.9 kB\u001b[0m \u001b[31m7.6 MB/s\u001b[0m eta \u001b[36m0:00:00\u001b[0m\n",
            "\u001b[?25h\u001b[?25l   \u001b[90m━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━\u001b[0m \u001b[32m0.0/66.8 kB\u001b[0m \u001b[31m?\u001b[0m eta \u001b[36m-:--:--\u001b[0m\r\u001b[2K   \u001b[90m━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━\u001b[0m \u001b[32m66.8/66.8 kB\u001b[0m \u001b[31m6.1 MB/s\u001b[0m eta \u001b[36m0:00:00\u001b[0m\n",
            "\u001b[2K   \u001b[90m━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━\u001b[0m \u001b[32m181.5/181.5 kB\u001b[0m \u001b[31m17.2 MB/s\u001b[0m eta \u001b[36m0:00:00\u001b[0m\n",
            "\u001b[2K   \u001b[90m━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━\u001b[0m \u001b[32m69.1/69.1 MB\u001b[0m \u001b[31m32.3 MB/s\u001b[0m eta \u001b[36m0:00:00\u001b[0m\n",
            "\u001b[?25h"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "!pip uninstall -y transformers"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "cZEiR39L_XvZ",
        "outputId": "c6b664bf-77e8-400c-b750-4a794a0282f6"
      },
      "execution_count": 3,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Found existing installation: transformers 4.46.3\n",
            "Uninstalling transformers-4.46.3:\n",
            "  Successfully uninstalled transformers-4.46.3\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "!pip install -q git+https://github.com/probicheaux/transformers.git@main"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "Ku_9wFMd_b9Q",
        "outputId": "ba9433e1-6bb9-410a-9a3a-fd563f64e35d"
      },
      "execution_count": 4,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "  Installing build dependencies ... \u001b[?25l\u001b[?25hdone\n",
            "  Getting requirements to build wheel ... \u001b[?25l\u001b[?25hdone\n",
            "  Preparing metadata (pyproject.toml) ... \u001b[?25l\u001b[?25hdone\n",
            "\u001b[2K   \u001b[90m━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━\u001b[0m \u001b[32m3.0/3.0 MB\u001b[0m \u001b[31m77.4 MB/s\u001b[0m eta \u001b[36m0:00:00\u001b[0m\n",
            "\u001b[?25h  Building wheel for transformers (pyproject.toml) ... \u001b[?25l\u001b[?25hdone\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 5,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "TGDFTYVnY4zn",
        "outputId": "d11fc59c-091c-4a22-ea92-4cb7cf7f7a8d"
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "loading Roboflow workspace...\n",
            "loading Roboflow project...\n"
          ]
        },
        {
          "output_type": "stream",
          "name": "stderr",
          "text": [
            "Downloading Dataset Version Zip in unsloth-latex-ocr-2 to jsonl:: 100%|██████████| 75703/75703 [00:05<00:00, 13110.13it/s]"
          ]
        },
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "\n"
          ]
        },
        {
          "output_type": "stream",
          "name": "stderr",
          "text": [
            "\n",
            "Extracting Dataset Version Zip to unsloth-latex-ocr-2 in jsonl:: 100%|██████████| 23947/23947 [00:02<00:00, 10276.47it/s]\n"
          ]
        }
      ],
      "source": [
        "from google.colab import userdata\n",
        "from roboflow import Roboflow\n",
        "\n",
        "ROBOFLOW_API_KEY = userdata.get('ROBOFLOW_API_KEY')\n",
        "rf = Roboflow(api_key=ROBOFLOW_API_KEY)\n",
        "\n",
        "project = rf.workspace(\"roboflow-jvuqo\").project(\"unsloth-latex-ocr\")\n",
        "version = project.version(2)\n",
        "dataset = version.download(\"jsonl\")"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "CU-Ol8tE61fI"
      },
      "source": [
        "**NOTE:** Let's read the first few lines of the annotation file and examine the dataset format."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 6,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "WLhSenP5AtQe",
        "outputId": "65f47ea2-9534-4997-f873-380f6e6f02bf"
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "{\"image\":\"0eb741ce399e127abe1d390aa7612979_png.rf.3399c69af6a1019db9095a1ac1327be8.jpg\",\"prefix\":\"read in latex\",\"suffix\":\"L = T _ { 1 0 } \\\\int d ^ { 1 0 } x [ 1 - ( - \\\\mathrm { D e t } ( \\\\eta _ { M N } + F _ { M N } ) ) ^ { { \\\\frac { 1 } { 2 } } } ]\"}\n",
            "{\"image\":\"2f3e2971f7cae6b0fc28b6ae6a58f3e2_png.rf.339a0574dace8f7a9bfad4083e558997.jpg\",\"prefix\":\"read in latex\",\"suffix\":\"\\\\Gamma ^ { a } : = \\\\gamma ^ { a } - i k ^ { a } \\\\gamma ^ { 3 } , \\\\qquad \\\\Gamma ^ { 3 } : = \\\\gamma ^ { 3 } + i k ^ { a } \\\\gamma _ { a } .\"}\n",
            "{\"image\":\"193bab6f414f17dbfce0cb14f891edc7_png.rf.33adb0be7b2bea8e03d7fd03420400ec.jpg\",\"prefix\":\"read in latex\",\"suffix\":\"\\\\Gamma [ h ] = \\\\frac { 1 } { 1 2 \\\\pi } t r \\\\int _ { B } d ^ { 3 } y \\\\epsilon _ { i j k } h ^ { - 1 } \\\\partial ^ { i } h h ^ { - 1 } \\\\partial ^ { j } h h ^ { - 1 } \\\\partial ^ { k } h ,\"}\n",
            "{\"image\":\"317dc5ccc58419fbeea610a84fd3d956_png.rf.33a4d3d873412bdeb07a8c4f6b453765.jpg\",\"prefix\":\"read in latex\",\"suffix\":\"\\\\delta W = { \\\\frac { m } { 2 } } \\\\mathrm { T r } \\\\Phi _ { 3 } ^ { 2 } .\"}\n",
            "{\"image\":\"e2732e11e0a3e3ae53459a29bb7cdbd2_png.rf.3393a9604c0747703957faa7292e3473.jpg\",\"prefix\":\"read in latex\",\"suffix\":\"N _ { 2 2 } ( x , y , a , b ) = \\\\frac { 1 } { 4 a b } \\\\sum _ { n = 1 } ^ { \\\\infty } \\\\cos ( \\\\frac { 2 n \\\\pi x } { a } ) \\\\int _ { \\\\frac { n \\\\pi } { a } b } ^ { \\\\infty } d q \\\\left( \\\\coth q - 1 \\\\right) \\\\cosh ( \\\\frac { 2 q y } { b } ) .\"}\n"
          ]
        }
      ],
      "source": [
        "!head -n 5 {dataset.location}/train/annotations.jsonl"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 7,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "PLehZP3xWCV8",
        "outputId": "531fe319-45b7-446f-e8ee-f022c3fba689"
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "19964\n",
            "1992\n",
            "1999\n"
          ]
        }
      ],
      "source": [
        "!wc -l < {dataset.location}/train/annotations.jsonl\n",
        "!wc -l < {dataset.location}/valid/annotations.jsonl\n",
        "!wc -l < {dataset.location}/test/annotations.jsonl"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# !head -n 10000 {dataset.location}/train/annotations.jsonl > {dataset.location}/train/annotations.sample.jsonl\n",
        "# !head -n 1000 {dataset.location}/valid/annotations.jsonl > {dataset.location}/valid/annotations.sample.jsonl\n",
        "# !head -n 1000 {dataset.location}/test/annotations.jsonl > {dataset.location}/test/annotations.sample.jsonl"
      ],
      "metadata": {
        "id": "2Q8_r99-BT7L"
      },
      "execution_count": 8,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "-8I_k2o77WI7"
      },
      "source": [
        "### Set up and test data loaders"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 9,
      "metadata": {
        "id": "R8ceYFDy7W3U"
      },
      "outputs": [],
      "source": [
        "import os\n",
        "import json\n",
        "import random\n",
        "from PIL import Image\n",
        "from torch.utils.data import Dataset\n",
        "\n",
        "\n",
        "class JSONLDataset(Dataset):\n",
        "    def __init__(self, jsonl_file_path: str, image_directory_path: str):\n",
        "        self.jsonl_file_path = jsonl_file_path\n",
        "        self.image_directory_path = image_directory_path\n",
        "        self.entries = self._load_entries()\n",
        "\n",
        "    def _load_entries(self):\n",
        "        entries = []\n",
        "        with open(self.jsonl_file_path, 'r') as file:\n",
        "            for line in file:\n",
        "                data = json.loads(line)\n",
        "                entries.append(data)\n",
        "        return entries\n",
        "\n",
        "    def __len__(self):\n",
        "        return len(self.entries)\n",
        "\n",
        "    def __getitem__(self, idx: int):\n",
        "        if idx < 0 or idx >= len(self.entries):\n",
        "            raise IndexError(\"Index out of range\")\n",
        "\n",
        "        entry = self.entries[idx]\n",
        "        image_path = os.path.join(self.image_directory_path, entry['image'])\n",
        "        image = Image.open(image_path)\n",
        "        return image, entry"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 10,
      "metadata": {
        "id": "lR4rf1Uu7bvc"
      },
      "outputs": [],
      "source": [
        "train_dataset = JSONLDataset(\n",
        "    jsonl_file_path=f\"{dataset.location}/train/annotations.jsonl\",\n",
        "    image_directory_path=f\"{dataset.location}/train\",\n",
        ")\n",
        "valid_dataset = JSONLDataset(\n",
        "    jsonl_file_path=f\"{dataset.location}/valid/annotations.jsonl\",\n",
        "    image_directory_path=f\"{dataset.location}/valid\",\n",
        ")\n",
        "test_dataset = JSONLDataset(\n",
        "    jsonl_file_path=f\"{dataset.location}/test/annotations.jsonl\",\n",
        "    image_directory_path=f\"{dataset.location}/test\",\n",
        ")"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 11,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 675
        },
        "id": "NCznuPbs87H6",
        "outputId": "9d34f13b-5bed-41a0-9920-0e717750b8e5"
      },
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<IPython.core.display.HTML object>"
            ],
            "text/html": [
              "\n",
              "    <html>\n",
              "    <head>\n",
              "        <style>\n",
              "            body { font-family: Arial, sans-serif; margin: 0; padding: 0; }\n",
              "            table { width: 100%; border-collapse: collapse; margin: 20px 0; }\n",
              "            th, td { border: 1px solid #ddd; padding: 8px; text-align: left; }\n",
              "            img { display: block; margin: auto; }\n",
              "        </style>\n",
              "    </head>\n",
              "    <body>\n",
              "        <table>\n",
              "            <tr>\n",
              "                <th>Image</th>\n",
              "                <th>Text</th>\n",
              "            </tr>\n",
              "            \n",
              "        <tr>\n",
              "            <td><img src=\"\" alt=\"Image\" style=\"max-width:300px; max-height:300px; object-fit:cover;\"></td>\n",
              "            <td>Prefix: read in latex<br>Suffix: L = T _ { 1 0 } \\int d ^ { 1 0 } x [ 1 - ( - \\mathrm { D e t } ( \\eta _ { M N } + F _ { M N } ) ) ^ { { \\frac { 1 } { 2 } } } ]</td>\n",
              "        </tr>\n",
              "        \n",
              "        <tr>\n",
              "            <td><img src=\"\" alt=\"Image\" style=\"max-width:300px; max-height:300px; object-fit:cover;\"></td>\n",
              "            <td>Prefix: read in latex<br>Suffix: \\Gamma ^ { a } : = \\gamma ^ { a } - i k ^ { a } \\gamma ^ { 3 } , \\qquad \\Gamma ^ { 3 } : = \\gamma ^ { 3 } + i k ^ { a } \\gamma _ { a } .</td>\n",
              "        </tr>\n",
              "        \n",
              "        <tr>\n",
              "            <td><img src=\"\" alt=\"Image\" style=\"max-width:300px; max-height:300px; object-fit:cover;\"></td>\n",
              "            <td>Prefix: read in latex<br>Suffix: \\Gamma [ h ] = \\frac { 1 } { 1 2 \\pi } t r \\int _ { B } d ^ { 3 } y \\epsilon _ { i j k } h ^ { - 1 } \\partial ^ { i } h h ^ { - 1 } \\partial ^ { j } h h ^ { - 1 } \\partial ^ { k } h ,</td>\n",
              "        </tr>\n",
              "        \n",
              "        <tr>\n",
              "            <td><img src=\"\" alt=\"Image\" style=\"max-width:300px; max-height:300px; object-fit:cover;\"></td>\n",
              "            <td>Prefix: read in latex<br>Suffix: \\delta W = { \\frac { m } { 2 } } \\mathrm { T r } \\Phi _ { 3 } ^ { 2 } .</td>\n",
              "        </tr>\n",
              "        \n",
              "        <tr>\n",
              "            <td><img src=\"\" alt=\"Image\" style=\"max-width:300px; max-height:300px; object-fit:cover;\"></td>\n",
              "            <td>Prefix: read in latex<br>Suffix: N _ { 2 2 } ( x , y , a , b ) = \\frac { 1 } { 4 a b } \\sum _ { n = 1 } ^ { \\infty } \\cos ( \\frac { 2 n \\pi x } { a } ) \\int _ { \\frac { n \\pi } { a } b } ^ { \\infty } d q \\left( \\coth q - 1 \\right) \\cosh ( \\frac { 2 q y } { b } ) .</td>\n",
              "        </tr>\n",
              "        \n",
              "        <tr>\n",
              "            <td><img src=\"\" alt=\"Image\" style=\"max-width:300px; max-height:300px; object-fit:cover;\"></td>\n",
              "            <td>Prefix: read in latex<br>Suffix: \\Phi = l \\ln g _ { s } , \\quad l = \\left( \\frac { 8 } { D - 2 } \\right) ^ { 1 / 2 } ,</td>\n",
              "        </tr>\n",
              "        \n",
              "        <tr>\n",
              "            <td><img src=\"\" alt=\"Image\" style=\"max-width:300px; max-height:300px; object-fit:cover;\"></td>\n",
              "            <td>Prefix: read in latex<br>Suffix: d s ^ { 2 } = d r ^ { 2 } + \\rho _ { 0 } ^ { 2 } \\sinh ^ { 2 } \\left( r / \\rho _ { 0 } \\right) \\left[ - H ^ { 2 } d t ^ { 2 } + \\cosh ^ { 2 } \\left( H t \\right) d \\Omega _ { 4 } ^ { 2 } \\right] .</td>\n",
              "        </tr>\n",
              "        \n",
              "        <tr>\n",
              "            <td><img src=\"\" alt=\"Image\" style=\"max-width:300px; max-height:300px; object-fit:cover;\"></td>\n",
              "            <td>Prefix: read in latex<br>Suffix: \\Sigma _ { \\mu \\nu } ( p , M ) = - \\epsilon _ { \\mu \\nu \\lambda } p _ { \\lambda }</td>\n",
              "        </tr>\n",
              "        \n",
              "        <tr>\n",
              "            <td><img src=\"\" alt=\"Image\" style=\"max-width:300px; max-height:300px; object-fit:cover;\"></td>\n",
              "            <td>Prefix: read in latex<br>Suffix: R _ { \\mu \\nu } = R _ { \\mu \\lambda \\nu } ^ { \\lambda } = \\partial _ { \\rho } \\gamma _ { \\mu \\nu } ^ { \\rho } - \\partial _ { \\nu } \\gamma _ { \\mu \\rho } ^ { \\rho } - \\gamma _ { \\mu \\lambda } ^ { \\rho } \\gamma _ { \\nu \\rho } ^ { \\lambda } + \\gamma _ { \\lambda \\rho } ^ { \\rho } \\gamma _ { \\mu \\nu } ^ { \\lambda } .</td>\n",
              "        </tr>\n",
              "        \n",
              "        <tr>\n",
              "            <td><img src=\"\" alt=\"Image\" style=\"max-width:300px; max-height:300px; object-fit:cover;\"></td>\n",
              "            <td>Prefix: read in latex<br>Suffix: \\Big [ { \\bf b } _ { i } ( \\vec { K } ) , { \\bf b } _ { i } ^ { \\dagger } ( { \\vec { K } } ^ { \\prime } ) \\Big ] = 2 ( 2 \\pi ) ^ { 3 } w _ { i } ( \\vec { K } ) \\delta ^ { 3 } ( \\vec { K } - { \\vec { K } } ^ { \\prime } ) ,</td>\n",
              "        </tr>\n",
              "        \n",
              "        </table>\n",
              "    </body>\n",
              "    </html>\n",
              "    "
            ]
          },
          "metadata": {}
        }
      ],
      "source": [
        "from IPython.core.display import display, HTML\n",
        "from PIL import Image\n",
        "import io\n",
        "import base64\n",
        "\n",
        "def pil_image_to_base64(img):\n",
        "    \"\"\"Convert a PIL image to a base64 string.\"\"\"\n",
        "    buffered = io.BytesIO()\n",
        "    img.save(buffered, format=\"JPEG\")\n",
        "    img_str = base64.b64encode(buffered.getvalue()).decode(\"utf-8\")\n",
        "    return f\"data:image/jpeg;base64,{img_str}\"\n",
        "\n",
        "def display_images_and_text(dataset, num_entries=10):\n",
        "    \"\"\"\n",
        "    Display images and their corresponding text side by side in an HTML table.\n",
        "\n",
        "    :param dataset: PyTorch dataset to extract images and texts from.\n",
        "    :param num_entries: Number of entries to display.\n",
        "    \"\"\"\n",
        "    images = []\n",
        "    texts = []\n",
        "\n",
        "    for i in range(min(num_entries, len(dataset))):\n",
        "        img, data = dataset[i]\n",
        "        images.append(pil_image_to_base64(img))\n",
        "        text = f\"Prefix: {data['prefix']}<br>Suffix: {data['suffix']}\"\n",
        "        texts.append(text)\n",
        "\n",
        "    rows = []\n",
        "    for img, text in zip(images, texts):\n",
        "        row_html = f\"\"\"\n",
        "        <tr>\n",
        "            <td><img src=\"{img}\" alt=\"Image\" style=\"max-width:300px; max-height:300px; object-fit:cover;\"></td>\n",
        "            <td>{text}</td>\n",
        "        </tr>\n",
        "        \"\"\"\n",
        "        rows.append(row_html)\n",
        "\n",
        "    html_content = f\"\"\"\n",
        "    <html>\n",
        "    <head>\n",
        "        <style>\n",
        "            body {{ font-family: Arial, sans-serif; margin: 0; padding: 0; }}\n",
        "            table {{ width: 100%; border-collapse: collapse; margin: 20px 0; }}\n",
        "            th, td {{ border: 1px solid #ddd; padding: 8px; text-align: left; }}\n",
        "            img {{ display: block; margin: auto; }}\n",
        "        </style>\n",
        "    </head>\n",
        "    <body>\n",
        "        <table>\n",
        "            <tr>\n",
        "                <th>Image</th>\n",
        "                <th>Text</th>\n",
        "            </tr>\n",
        "            {''.join(rows)}\n",
        "        </table>\n",
        "    </body>\n",
        "    </html>\n",
        "    \"\"\"\n",
        "\n",
        "    display(HTML(html_content))\n",
        "\n",
        "display_images_and_text(train_dataset, num_entries=10)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "_ZvYNxYbBtE3"
      },
      "source": [
        "### Load PaliGemma2 model\n",
        "\n",
        "**NOTE:** PaliGemma2 offers 9 pre-trained models with sizes of `3B`, `10B`, and `28B` parameters, and resolutions of `224`, `448`, and `896` pixels. In this tutorial, I'll be using the [`google/paligemma2-10b-pt-448`](https://huggingface.co/google/paligemma2-10b-pt-224) checkpoint."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 12,
      "metadata": {
        "id": "l0DpE_ibx9aB"
      },
      "outputs": [],
      "source": [
        "import torch\n",
        "from transformers import PaliGemmaProcessor, PaliGemmaForConditionalGeneration\n",
        "\n",
        "MODEL_ID =\"google/paligemma2-10b-pt-224\"\n",
        "DEVICE = torch.device(\"cuda\" if torch.cuda.is_available() else \"cpu\")"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 13,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 303,
          "referenced_widgets": [
            "f84389f96f5b4f0083393a909405d361",
            "0a2efe8e7cc0441bb245a16143fccacd",
            "8efc673032d94642803c17bca21be187",
            "94d4353a335d4b27b0279db0ee02949a",
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            "3e218b0a4a34445080a5a2ed3eef511b",
            "60e707662cdd4337a611ec62a7c0903d",
            "f6372b23105b49ab8ade7802c5c8064a",
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            "654e31910dbb47cfb9a544dc6c5f000c",
            "ae5f08a1ac7f421f91ca2e64a44a6b2b"
          ]
        },
        "id": "_DgILBz3LHw1",
        "outputId": "4733164c-e9b2-4027-e033-3db22b9bf1f3"
      },
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "VBox(children=(HTML(value='<center> <img\\nsrc=https://huggingface.co/front/assets/huggingface_logo-noborder.sv…"
            ],
            "application/vnd.jupyter.widget-view+json": {
              "version_major": 2,
              "version_minor": 0,
              "model_id": "f84389f96f5b4f0083393a909405d361"
            }
          },
          "metadata": {}
        }
      ],
      "source": [
        "from huggingface_hub import notebook_login\n",
        "notebook_login()"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 14,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 145,
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        },
        "id": "ntXj4A3SyEAa",
        "outputId": "f924b914-b394-4e74-c786-5178895f582d"
      },
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "preprocessor_config.json:   0%|          | 0.00/424 [00:00<?, ?B/s]"
            ],
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          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "tokenizer_config.json:   0%|          | 0.00/243k [00:00<?, ?B/s]"
            ],
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              "version_minor": 0,
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          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "tokenizer.json:   0%|          | 0.00/34.6M [00:00<?, ?B/s]"
            ],
            "application/vnd.jupyter.widget-view+json": {
              "version_major": 2,
              "version_minor": 0,
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            }
          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "special_tokens_map.json:   0%|          | 0.00/733 [00:00<?, ?B/s]"
            ],
            "application/vnd.jupyter.widget-view+json": {
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              "model_id": "8491ed88a8034410b0def84abf09f250"
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          },
          "metadata": {}
        }
      ],
      "source": [
        "processor = PaliGemmaProcessor.from_pretrained(MODEL_ID)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 15,
      "metadata": {
        "id": "_qsRTZZAM03R"
      },
      "outputs": [],
      "source": [
        "USE_LORA = False\n",
        "USE_QLORA = True\n",
        "FREEZE_VISION = False"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 16,
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        "colab": {
          "base_uri": "https://localhost:8080/",
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        "id": "rEpM9zgOM5lr",
        "outputId": "34ef8b04-5cdd-4f84-86d5-f0167233d7a9"
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stderr",
          "text": [
            "Unused kwargs: ['bnb_4bit_compute_type']. These kwargs are not used in <class 'transformers.utils.quantization_config.BitsAndBytesConfig'>.\n"
          ]
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "config.json:   0%|          | 0.00/1.33k [00:00<?, ?B/s]"
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          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "model.safetensors.index.json:   0%|          | 0.00/92.6k [00:00<?, ?B/s]"
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        {
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          "data": {
            "text/plain": [
              "Downloading shards:   0%|          | 0/4 [00:00<?, ?it/s]"
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          },
          "metadata": {}
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            }
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          "metadata": {}
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          },
          "metadata": {}
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        {
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          "data": {
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            ],
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              "version_major": 2,
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            }
          },
          "metadata": {}
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            ],
            "application/vnd.jupyter.widget-view+json": {
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            }
          },
          "metadata": {}
        },
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "trainable params: 28,502,016 || all params: 9,691,140,848 || trainable%: 0.2941\n"
          ]
        }
      ],
      "source": [
        "from peft import get_peft_model, LoraConfig\n",
        "from transformers import BitsAndBytesConfig\n",
        "\n",
        "if USE_LORA or USE_QLORA:\n",
        "    lora_config = LoraConfig(\n",
        "    r=8,\n",
        "    target_modules=[\"q_proj\", \"o_proj\", \"k_proj\", \"v_proj\", \"gate_proj\", \"up_proj\", \"down_proj\"],\n",
        "    task_type=\"CAUSAL_LM\",\n",
        "    )\n",
        "    if USE_QLORA:\n",
        "        bnb_config = BitsAndBytesConfig(\n",
        "        load_in_4bit=True,\n",
        "        bnb_4bit_quant_type=\"nf4\",\n",
        "        bnb_4bit_compute_type=torch.bfloat16\n",
        "            )\n",
        "    model = PaliGemmaForConditionalGeneration.from_pretrained(\n",
        "        MODEL_ID,\n",
        "        device_map=\"auto\",\n",
        "        quantization_config=bnb_config if USE_QLORA else None,\n",
        "        torch_dtype=torch.bfloat16)\n",
        "    model = get_peft_model(model, lora_config)\n",
        "    model = model.to(DEVICE)\n",
        "    model.print_trainable_parameters()\n",
        "else:\n",
        "    model = PaliGemmaForConditionalGeneration.from_pretrained(MODEL_ID, device_map=\"auto\").to(DEVICE)\n",
        "    model = model.to(DEVICE)\n",
        "\n",
        "    if FREEZE_VISION:\n",
        "        for param in model.vision_tower.parameters():\n",
        "            param.requires_grad = False\n",
        "\n",
        "        for param in model.multi_modal_projector.parameters():\n",
        "            param.requires_grad = False\n",
        "\n",
        "TORCH_DTYPE = model.dtype"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "8W1JY6x98WTq"
      },
      "source": [
        "### Fine-tune PaliGemma2 on LaTeX OCR dataset"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 17,
      "metadata": {
        "id": "ErMInlMt8gHU"
      },
      "outputs": [],
      "source": [
        "from transformers import Trainer, TrainingArguments\n",
        "\n",
        "\n",
        "def collate_fn(batch):\n",
        "    images, labels = zip(*batch)\n",
        "\n",
        "    paths = [label[\"image\"] for label in labels]\n",
        "    prefixes = [\"<image>\" + label[\"prefix\"] for label in labels]\n",
        "    suffixes = [label[\"suffix\"] for label in labels]\n",
        "\n",
        "    inputs = processor(\n",
        "        text=prefixes,\n",
        "        images=images,\n",
        "        return_tensors=\"pt\",\n",
        "        suffix=suffixes,\n",
        "        padding=\"longest\"\n",
        "    ).to(TORCH_DTYPE).to(DEVICE)\n",
        "\n",
        "    return inputs\n",
        "\n",
        "args = TrainingArguments(\n",
        "    num_train_epochs=3,\n",
        "    remove_unused_columns=False,\n",
        "    per_device_train_batch_size=3,\n",
        "    gradient_accumulation_steps=12,\n",
        "    warmup_steps=2,\n",
        "    learning_rate=2e-5,\n",
        "    weight_decay=1e-6,\n",
        "    adam_beta2=0.999,\n",
        "    logging_steps=100,\n",
        "    optim=\"paged_adamw_8bit\" if USE_QLORA else \"adamw_hf\",\n",
        "    save_strategy=\"steps\",\n",
        "    save_steps=1000,\n",
        "    save_total_limit=1,\n",
        "    output_dir=\"paligemma2_latex_ocr_v5\",\n",
        "    bf16=True,\n",
        "    report_to=[\"tensorboard\"],\n",
        "    dataloader_pin_memory=False\n",
        ")\n",
        "\n",
        "trainer = Trainer(\n",
        "    model=model,\n",
        "    train_dataset=train_dataset,\n",
        "    eval_dataset=valid_dataset,\n",
        "    data_collator=collate_fn,\n",
        "    args=args\n",
        ")"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 18,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 594
        },
        "id": "6T8G03yQ9LJF",
        "outputId": "6b7b32b1-3132-46cf-d737-f77673b57126"
      },
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<IPython.core.display.HTML object>"
            ],
            "text/html": [
              "\n",
              "    <div>\n",
              "      \n",
              "      <progress value='1662' max='1662' style='width:300px; height:20px; vertical-align: middle;'></progress>\n",
              "      [1662/1662 4:15:51, Epoch 2/3]\n",
              "    </div>\n",
              "    <table border=\"1\" class=\"dataframe\">\n",
              "  <thead>\n",
              " <tr style=\"text-align: left;\">\n",
              "      <th>Step</th>\n",
              "      <th>Training Loss</th>\n",
              "    </tr>\n",
              "  </thead>\n",
              "  <tbody>\n",
              "    <tr>\n",
              "      <td>100</td>\n",
              "      <td>9.844200</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <td>200</td>\n",
              "      <td>5.440100</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <td>300</td>\n",
              "      <td>4.329200</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <td>400</td>\n",
              "      <td>3.641700</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <td>500</td>\n",
              "      <td>3.248500</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <td>600</td>\n",
              "      <td>2.916000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <td>700</td>\n",
              "      <td>2.652100</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <td>800</td>\n",
              "      <td>2.485000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <td>900</td>\n",
              "      <td>2.333900</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <td>1000</td>\n",
              "      <td>2.345000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <td>1100</td>\n",
              "      <td>2.157600</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <td>1200</td>\n",
              "      <td>2.009000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <td>1300</td>\n",
              "      <td>1.928500</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <td>1400</td>\n",
              "      <td>1.859700</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <td>1500</td>\n",
              "      <td>1.915800</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <td>1600</td>\n",
              "      <td>1.883600</td>\n",
              "    </tr>\n",
              "  </tbody>\n",
              "</table><p>"
            ]
          },
          "metadata": {}
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "TrainOutput(global_step=1662, training_loss=3.1394706432282855, metrics={'train_runtime': 15363.293, 'train_samples_per_second': 3.899, 'train_steps_per_second': 0.108, 'total_flos': 1.1381261608321336e+18, 'train_loss': 3.1394706432282855, 'epoch': 2.99534184823441})"
            ]
          },
          "metadata": {},
          "execution_count": 18
        }
      ],
      "source": [
        "trainer.train()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "1c4FK0XXCxPU"
      },
      "source": [
        "### Run inference with fine-tuned PaliGemma2 model"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 19,
      "metadata": {
        "id": "3a8sis7pHGtM"
      },
      "outputs": [],
      "source": [
        "# @title Function to render text diffs\n",
        "\n",
        "from difflib import SequenceMatcher\n",
        "from IPython.core.display import display, HTML\n",
        "\n",
        "def side_by_side_diff_divs(text1, text2):\n",
        "    lines1 = text1.splitlines()\n",
        "    lines2 = text2.splitlines()\n",
        "\n",
        "    original_output = []\n",
        "    modified_output = []\n",
        "\n",
        "    for line1, line2 in zip(lines1, lines2):\n",
        "        words1 = line1.split()\n",
        "        words2 = line2.split()\n",
        "\n",
        "        matcher = SequenceMatcher(None, words1, words2)\n",
        "\n",
        "        original_line = []\n",
        "        modified_line = []\n",
        "\n",
        "        for tag, i1, i2, j1, j2 in matcher.get_opcodes():\n",
        "            if tag == 'replace':\n",
        "                original_line.append(f\"<span class='diff-remove'>{' '.join(words1[i1:i2])}</span>\")\n",
        "                modified_line.append(f\"<span class='diff-add'>{' '.join(words2[j1:j2])}</span>\")\n",
        "            elif tag == 'delete':\n",
        "                original_line.append(f\"<span class='diff-remove'>{' '.join(words1[i1:i2])}</span>\")\n",
        "            elif tag == 'insert':\n",
        "                modified_line.append(f\"<span class='diff-add'>{' '.join(words2[j1:j2])}</span>\")\n",
        "            elif tag == 'equal':\n",
        "                original_line.append(' '.join(words1[i1:i2]))\n",
        "                modified_line.append(' '.join(words2[j1:j2]))\n",
        "\n",
        "        original_output.append(' '.join(original_line) + \"<br>\")\n",
        "        modified_output.append(' '.join(modified_line) + \"<br>\")\n",
        "\n",
        "    original_html = \"<br>\" + ''.join(original_output) + \"<br>\"\n",
        "    modified_html = \"<br>\" + ''.join(modified_output) + \"<br>\"\n",
        "\n",
        "    html = f\"\"\"\n",
        "    <html>\n",
        "    <head>\n",
        "        <style>\n",
        "            body {{ font-family: Arial, sans-serif; margin: 0; padding: 0; }}\n",
        "            .container {{ display: flex; align-items: flex-start; }}\n",
        "            .column {{\n",
        "                flex: 1;\n",
        "                padding: 10px;\n",
        "                white-space: pre-wrap;\n",
        "                text-align: left;\n",
        "            }}\n",
        "            .diff-remove {{\n",
        "                background-color: #d9534f;  /* Dark red */\n",
        "                color: white;\n",
        "                text-decoration: line-through;\n",
        "                border-radius: 4px;\n",
        "                padding: 2px 4px;\n",
        "            }}\n",
        "            .diff-add {{\n",
        "                background-color: #5cb85c;  /* Dark green */\n",
        "                color: white;\n",
        "                border-radius: 4px;\n",
        "                padding: 2px 4px;\n",
        "            }}\n",
        "        </style>\n",
        "    </head>\n",
        "    <body>\n",
        "        <div class=\"container\">\n",
        "            <div class=\"column\" style=\"border-right: 1px solid #ccc;\">\n",
        "                {original_html}\n",
        "            </div>\n",
        "            <div class=\"column\">\n",
        "                {modified_html}\n",
        "            </div>\n",
        "        </div>\n",
        "    </body>\n",
        "    </html>\n",
        "    \"\"\"\n",
        "    return html"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 20,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 1000
        },
        "id": "dRR4I4QANy5K",
        "outputId": "b594b0e1-0c5a-4f18-9fb9-1701cac8c047"
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stderr",
          "text": [
            "The 'batch_size' attribute of HybridCache is deprecated and will be removed in v4.49. Use the more precisely named 'self.max_batch_size' attribute instead.\n"
          ]
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<PIL.JpegImagePlugin.JpegImageFile image mode=RGB size=280x40>"
            ],
            "image/png": 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\n",
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          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<IPython.core.display.HTML object>"
            ],
            "text/html": [
              "\n",
              "    <html>\n",
              "    <head>\n",
              "        <style>\n",
              "            body { font-family: Arial, sans-serif; margin: 0; padding: 0; }\n",
              "            .container { display: flex; align-items: flex-start; }\n",
              "            .column {\n",
              "                flex: 1;\n",
              "                padding: 10px;\n",
              "                white-space: pre-wrap;\n",
              "                text-align: left;\n",
              "            }\n",
              "            .diff-remove {\n",
              "                background-color: #d9534f;  /* Dark red */\n",
              "                color: white;\n",
              "                text-decoration: line-through;\n",
              "                border-radius: 4px;\n",
              "                padding: 2px 4px;\n",
              "            }\n",
              "            .diff-add {\n",
              "                background-color: #5cb85c;  /* Dark green */\n",
              "                color: white;\n",
              "                border-radius: 4px;\n",
              "                padding: 2px 4px;\n",
              "            }\n",
              "        </style>\n",
              "    </head>\n",
              "    <body>\n",
              "        <div class=\"container\">\n",
              "            <div class=\"column\" style=\"border-right: 1px solid #ccc;\">\n",
              "                <br>H = <span class='diff-remove'>\\dot {</span> x <span class='diff-remove'>}</span> _ { i } \\Pi _ { x <span class='diff-remove'>^ { i }</span> } + \\Pi _ { x <span class='diff-remove'>^ { i</span> } <span class='diff-remove'>} \\dot {</span> x <span class='diff-remove'>}</span> _ { i } ^ { <span class='diff-remove'>*</span> } + <span class='diff-remove'>\\dot {</span> \\psi <span class='diff-remove'>} _ { i }</span> \\Pi _ { \\psi <span class='diff-remove'>_ { i }</span> } - \\Pi _ { \\psi <span class='diff-remove'>_ { i</span> } ^ { <span class='diff-remove'>* } } \\dot { \\psi } _ { i } ^ { *</span> } +<br><br>\n",
              "            </div>\n",
              "            <div class=\"column\">\n",
              "                <br>H = x _ { i } \\Pi _ { x } + \\Pi _ { x } x _ { i } ^ { <span class='diff-add'>\\ast</span> } + <span class='diff-add'>i</span> \\psi \\Pi _ { \\psi } - \\Pi _ { \\psi } <span class='diff-add'>\\psi</span> ^ { <span class='diff-add'>\\ast</span> } +<br><br>\n",
              "            </div>\n",
              "        </div>\n",
              "    </body>\n",
              "    </html>\n",
              "    "
            ]
          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<PIL.JpegImagePlugin.JpegImageFile image mode=RGB size=360x40>"
            ],
            "image/png": 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        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<IPython.core.display.HTML object>"
            ],
            "text/html": [
              "\n",
              "    <html>\n",
              "    <head>\n",
              "        <style>\n",
              "            body { font-family: Arial, sans-serif; margin: 0; padding: 0; }\n",
              "            .container { display: flex; align-items: flex-start; }\n",
              "            .column {\n",
              "                flex: 1;\n",
              "                padding: 10px;\n",
              "                white-space: pre-wrap;\n",
              "                text-align: left;\n",
              "            }\n",
              "            .diff-remove {\n",
              "                background-color: #d9534f;  /* Dark red */\n",
              "                color: white;\n",
              "                text-decoration: line-through;\n",
              "                border-radius: 4px;\n",
              "                padding: 2px 4px;\n",
              "            }\n",
              "            .diff-add {\n",
              "                background-color: #5cb85c;  /* Dark green */\n",
              "                color: white;\n",
              "                border-radius: 4px;\n",
              "                padding: 2px 4px;\n",
              "            }\n",
              "        </style>\n",
              "    </head>\n",
              "    <body>\n",
              "        <div class=\"container\">\n",
              "            <div class=\"column\" style=\"border-right: 1px solid #ccc;\">\n",
              "                <br>\\psi _ { j } ( C _ { r } <span class='diff-remove'>^ { \\vee }</span> , t ) = \\frac { 4 \\sinh 2 j <span class='diff-remove'>t</span> ( \\cosh ( 2 <span class='diff-remove'>w</span> _ { 1 } t ) \\cosh ( 2 <span class='diff-remove'>w</span> _ { 2 } t ) - \\cos ^ { 2 } ( x t ) ) } { \\sinh 2 t \\cosh <span class='diff-remove'>h</span> t } .<br><br>\n",
              "            </div>\n",
              "            <div class=\"column\">\n",
              "                <br>\\psi _ { j } ( C _ { r } , t ) = \\frac { 4 \\sinh 2 j ( \\cosh ( 2 <span class='diff-add'>u</span> _ { 1 } t ) \\cosh ( 2 <span class='diff-add'>u</span> _ { 2 } t ) - \\cos ^ { 2 } ( x t ) ) } { \\sinh 2 t \\cosh <span class='diff-add'>k</span> t } .<br><br>\n",
              "            </div>\n",
              "        </div>\n",
              "    </body>\n",
              "    </html>\n",
              "    "
            ]
          },
          "metadata": {}
        },
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          "output_type": "display_data",
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            ],
            "image/png": 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\n",
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          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<IPython.core.display.HTML object>"
            ],
            "text/html": [
              "\n",
              "    <html>\n",
              "    <head>\n",
              "        <style>\n",
              "            body { font-family: Arial, sans-serif; margin: 0; padding: 0; }\n",
              "            .container { display: flex; align-items: flex-start; }\n",
              "            .column {\n",
              "                flex: 1;\n",
              "                padding: 10px;\n",
              "                white-space: pre-wrap;\n",
              "                text-align: left;\n",
              "            }\n",
              "            .diff-remove {\n",
              "                background-color: #d9534f;  /* Dark red */\n",
              "                color: white;\n",
              "                text-decoration: line-through;\n",
              "                border-radius: 4px;\n",
              "                padding: 2px 4px;\n",
              "            }\n",
              "            .diff-add {\n",
              "                background-color: #5cb85c;  /* Dark green */\n",
              "                color: white;\n",
              "                border-radius: 4px;\n",
              "                padding: 2px 4px;\n",
              "            }\n",
              "        </style>\n",
              "    </head>\n",
              "    <body>\n",
              "        <div class=\"container\">\n",
              "            <div class=\"column\" style=\"border-right: 1px solid #ccc;\">\n",
              "                <br><span class='diff-remove'>-</span> \\frac { <span class='diff-remove'>h</span> ^ { 2 } } { 2 <span class='diff-remove'>\\lambda</span> } \\int d <span class='diff-remove'>t d</span> ^ { <span class='diff-remove'>2</span> } x <span class='diff-remove'>d</span> ^ { <span class='diff-remove'>2</span> } x ^ { <span class='diff-remove'>\\prime</span> } ( <span class='diff-remove'>{ \\tilde {</span> J <span class='diff-remove'>} }</span> _ { k } - \\frac { <span class='diff-remove'>J</span> _ { <span class='diff-remove'>k</span> } ^ { <span class='diff-remove'>0</span> } } { \\rho _ { 0 } } <span class='diff-remove'>{ \\tilde</span> { J <span class='diff-remove'>}</span> } _ { 0 } ) ( <span class='diff-remove'>t , x )</span> \\Delta ^ { - 1 } ( x - x ^ { \\prime } ) <span class='diff-remove'>( { \\tilde { J } } _ { k } - \\frac {</span> J _ { k } ^ { <span class='diff-remove'>0</span> } } { \\rho _ { 0 } } <span class='diff-remove'>{ \\tilde</span> { J <span class='diff-remove'>}</span> } _ { 0 } ) ( <span class='diff-remove'>t</span> , x ^ { \\prime } ) <span class='diff-remove'>.</span><br><br>\n",
              "            </div>\n",
              "            <div class=\"column\">\n",
              "                <br>\\frac { <span class='diff-add'>k</span> ^ { 2 } } { 2 <span class='diff-add'>k</span> } \\int d <span class='diff-add'>x</span> ^ { <span class='diff-add'>0</span> } <span class='diff-add'>d</span> x ^ { <span class='diff-add'>1</span> } x ^ { <span class='diff-add'>0</span> } ( J _ { k } - \\frac { <span class='diff-add'>\\mu</span> _ { <span class='diff-add'>0</span> } ^ { <span class='diff-add'>2</span> } } { \\rho _ { 0 } } <span class='diff-add'>\\bar</span> { J } _ { 0 } ) ( <span class='diff-add'>1 -</span> \\Delta ^ { - 1 } ( x - x ^ { \\prime } ) <span class='diff-add'>)</span> J _ { k } <span class='diff-add'>( - \\frac { \\mu _ { 0 }</span> ^ { <span class='diff-add'>2</span> } } { \\rho _ { 0 } } <span class='diff-add'>\\bar</span> { J } _ { 0 } ) ( <span class='diff-add'>k</span> , x ^ { \\prime } )<br><br>\n",
              "            </div>\n",
              "        </div>\n",
              "    </body>\n",
              "    </html>\n",
              "    "
            ]
          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
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              "<PIL.JpegImagePlugin.JpegImageFile image mode=RGB size=160x40>"
            ],
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\n",
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          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<IPython.core.display.HTML object>"
            ],
            "text/html": [
              "\n",
              "    <html>\n",
              "    <head>\n",
              "        <style>\n",
              "            body { font-family: Arial, sans-serif; margin: 0; padding: 0; }\n",
              "            .container { display: flex; align-items: flex-start; }\n",
              "            .column {\n",
              "                flex: 1;\n",
              "                padding: 10px;\n",
              "                white-space: pre-wrap;\n",
              "                text-align: left;\n",
              "            }\n",
              "            .diff-remove {\n",
              "                background-color: #d9534f;  /* Dark red */\n",
              "                color: white;\n",
              "                text-decoration: line-through;\n",
              "                border-radius: 4px;\n",
              "                padding: 2px 4px;\n",
              "            }\n",
              "            .diff-add {\n",
              "                background-color: #5cb85c;  /* Dark green */\n",
              "                color: white;\n",
              "                border-radius: 4px;\n",
              "                padding: 2px 4px;\n",
              "            }\n",
              "        </style>\n",
              "    </head>\n",
              "    <body>\n",
              "        <div class=\"container\">\n",
              "            <div class=\"column\" style=\"border-right: 1px solid #ccc;\">\n",
              "                <br>\\bar { \\Phi } _ { 0 } = \\frac { ( \\tilde { Q } _ { 0 } \\tilde { Q } _ { 1 } \\tilde { Q } _ { 2 } ) ^ { 1 / 2 } } { l ^ { 3 } }<br><br>\n",
              "            </div>\n",
              "            <div class=\"column\">\n",
              "                <br>\\bar { \\Phi } _ { 0 } = \\frac { ( \\tilde { Q } _ { 0 } \\tilde { Q } _ { 1 } \\tilde { Q } _ { 2 } ) ^ { 1 / 2 } } { l ^ { 3 } }<br><br>\n",
              "            </div>\n",
              "        </div>\n",
              "    </body>\n",
              "    </html>\n",
              "    "
            ]
          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<PIL.JpegImagePlugin.JpegImageFile image mode=RGB size=200x50>"
            ],
            "image/png": 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\n",
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          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<IPython.core.display.HTML object>"
            ],
            "text/html": [
              "\n",
              "    <html>\n",
              "    <head>\n",
              "        <style>\n",
              "            body { font-family: Arial, sans-serif; margin: 0; padding: 0; }\n",
              "            .container { display: flex; align-items: flex-start; }\n",
              "            .column {\n",
              "                flex: 1;\n",
              "                padding: 10px;\n",
              "                white-space: pre-wrap;\n",
              "                text-align: left;\n",
              "            }\n",
              "            .diff-remove {\n",
              "                background-color: #d9534f;  /* Dark red */\n",
              "                color: white;\n",
              "                text-decoration: line-through;\n",
              "                border-radius: 4px;\n",
              "                padding: 2px 4px;\n",
              "            }\n",
              "            .diff-add {\n",
              "                background-color: #5cb85c;  /* Dark green */\n",
              "                color: white;\n",
              "                border-radius: 4px;\n",
              "                padding: 2px 4px;\n",
              "            }\n",
              "        </style>\n",
              "    </head>\n",
              "    <body>\n",
              "        <div class=\"container\">\n",
              "            <div class=\"column\" style=\"border-right: 1px solid #ccc;\">\n",
              "                <br>f ( \\sigma ) = f _ { 0 } \\frac { 1 } { \\sqrt { \\pi } } + \\sum _ { <span class='diff-remove'>r</span> = 1 } ^ { \\infty } f _ { n }<br><br>\n",
              "            </div>\n",
              "            <div class=\"column\">\n",
              "                <br>f ( \\sigma ) = f _ { 0 } \\frac { 1 } { \\sqrt { \\pi } } + \\sum _ { <span class='diff-add'>n</span> = 1 } ^ { \\infty } f _ { n }<br><br>\n",
              "            </div>\n",
              "        </div>\n",
              "    </body>\n",
              "    </html>\n",
              "    "
            ]
          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<PIL.JpegImagePlugin.JpegImageFile image mode=RGB size=200x40>"
            ],
            "image/png": 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\n",
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          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<IPython.core.display.HTML object>"
            ],
            "text/html": [
              "\n",
              "    <html>\n",
              "    <head>\n",
              "        <style>\n",
              "            body { font-family: Arial, sans-serif; margin: 0; padding: 0; }\n",
              "            .container { display: flex; align-items: flex-start; }\n",
              "            .column {\n",
              "                flex: 1;\n",
              "                padding: 10px;\n",
              "                white-space: pre-wrap;\n",
              "                text-align: left;\n",
              "            }\n",
              "            .diff-remove {\n",
              "                background-color: #d9534f;  /* Dark red */\n",
              "                color: white;\n",
              "                text-decoration: line-through;\n",
              "                border-radius: 4px;\n",
              "                padding: 2px 4px;\n",
              "            }\n",
              "            .diff-add {\n",
              "                background-color: #5cb85c;  /* Dark green */\n",
              "                color: white;\n",
              "                border-radius: 4px;\n",
              "                padding: 2px 4px;\n",
              "            }\n",
              "        </style>\n",
              "    </head>\n",
              "    <body>\n",
              "        <div class=\"container\">\n",
              "            <div class=\"column\" style=\"border-right: 1px solid #ccc;\">\n",
              "                <br>C _ { \\mathrm { b u l k } } ^ { ( 3 ) } = \\frac { 2 \\sqrt { 2 } } { 3 } \\beta \\sinh ( \\sqrt { 2 } \\beta \\phi _ { 0 } ) ,<br><br>\n",
              "            </div>\n",
              "            <div class=\"column\">\n",
              "                <br>C _ { \\mathrm { b u l k } } ^ { ( 3 ) } = \\frac { 2 \\sqrt { 2 } } { 3 } \\beta \\sinh ( \\sqrt { 2 } \\beta \\phi _ { 0 } ) ,<br><br>\n",
              "            </div>\n",
              "        </div>\n",
              "    </body>\n",
              "    </html>\n",
              "    "
            ]
          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
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              "<PIL.JpegImagePlugin.JpegImageFile image mode=RGB size=200x50>"
            ],
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          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<IPython.core.display.HTML object>"
            ],
            "text/html": [
              "\n",
              "    <html>\n",
              "    <head>\n",
              "        <style>\n",
              "            body { font-family: Arial, sans-serif; margin: 0; padding: 0; }\n",
              "            .container { display: flex; align-items: flex-start; }\n",
              "            .column {\n",
              "                flex: 1;\n",
              "                padding: 10px;\n",
              "                white-space: pre-wrap;\n",
              "                text-align: left;\n",
              "            }\n",
              "            .diff-remove {\n",
              "                background-color: #d9534f;  /* Dark red */\n",
              "                color: white;\n",
              "                text-decoration: line-through;\n",
              "                border-radius: 4px;\n",
              "                padding: 2px 4px;\n",
              "            }\n",
              "            .diff-add {\n",
              "                background-color: #5cb85c;  /* Dark green */\n",
              "                color: white;\n",
              "                border-radius: 4px;\n",
              "                padding: 2px 4px;\n",
              "            }\n",
              "        </style>\n",
              "    </head>\n",
              "    <body>\n",
              "        <div class=\"container\">\n",
              "            <div class=\"column\" style=\"border-right: 1px solid #ccc;\">\n",
              "                <br>\\pi ^ { \\mu } ( x ) = \\frac { \\partial { \\cal <span class='diff-remove'>{</span> L <span class='diff-remove'>}</span> } } { \\partial \\dot { A _ { \\mu <span class='diff-remove'>}</span> } } = - F ^ { 0 \\mu } ( x ) .<br><br>\n",
              "            </div>\n",
              "            <div class=\"column\">\n",
              "                <br>\\pi ^ { \\mu } ( x ) = \\frac { \\partial { \\cal L } } { \\partial \\dot { A <span class='diff-add'>}</span> _ { \\mu } } = - F ^ { 0 \\mu } ( x ) .<br><br>\n",
              "            </div>\n",
              "        </div>\n",
              "    </body>\n",
              "    </html>\n",
              "    "
            ]
          },
          "metadata": {}
        },
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          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<PIL.JpegImagePlugin.JpegImageFile image mode=RGB size=320x40>"
            ],
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\n",
            "image/jpeg": "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\n"
          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<IPython.core.display.HTML object>"
            ],
            "text/html": [
              "\n",
              "    <html>\n",
              "    <head>\n",
              "        <style>\n",
              "            body { font-family: Arial, sans-serif; margin: 0; padding: 0; }\n",
              "            .container { display: flex; align-items: flex-start; }\n",
              "            .column {\n",
              "                flex: 1;\n",
              "                padding: 10px;\n",
              "                white-space: pre-wrap;\n",
              "                text-align: left;\n",
              "            }\n",
              "            .diff-remove {\n",
              "                background-color: #d9534f;  /* Dark red */\n",
              "                color: white;\n",
              "                text-decoration: line-through;\n",
              "                border-radius: 4px;\n",
              "                padding: 2px 4px;\n",
              "            }\n",
              "            .diff-add {\n",
              "                background-color: #5cb85c;  /* Dark green */\n",
              "                color: white;\n",
              "                border-radius: 4px;\n",
              "                padding: 2px 4px;\n",
              "            }\n",
              "        </style>\n",
              "    </head>\n",
              "    <body>\n",
              "        <div class=\"container\">\n",
              "            <div class=\"column\" style=\"border-right: 1px solid #ccc;\">\n",
              "                <br><span class='diff-remove'>{ \\cal</span> P <span class='diff-remove'>}</span> Q ^ { ( <span class='diff-remove'>a</span> ) } | E _ { n } \\rangle = <span class='diff-remove'>{ \\cal</span> P <span class='diff-remove'>}</span> ( - i ) D ^ { ( <span class='diff-remove'>a</span> ) } ( X _ { n } ) | 0 \\rangle = - P <span class='diff-remove'>_ { n }</span> Q ^ { ( <span class='diff-remove'>a</span> ) } | E _ { n } \\rangle ,<br><br>\n",
              "            </div>\n",
              "            <div class=\"column\">\n",
              "                <br>P Q ^ { ( <span class='diff-add'>s</span> ) } | E _ { n } \\rangle = P ( - i ) D ^ { ( <span class='diff-add'>s</span> ) } ( X _ { n } ) | 0 \\rangle = - P <span class='diff-add'>\\</span> Q ^ { ( <span class='diff-add'>s</span> ) } | E _ { n } \\rangle ,<br><br>\n",
              "            </div>\n",
              "        </div>\n",
              "    </body>\n",
              "    </html>\n",
              "    "
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          "metadata": {}
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          "output_type": "display_data",
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\n",
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          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<IPython.core.display.HTML object>"
            ],
            "text/html": [
              "\n",
              "    <html>\n",
              "    <head>\n",
              "        <style>\n",
              "            body { font-family: Arial, sans-serif; margin: 0; padding: 0; }\n",
              "            .container { display: flex; align-items: flex-start; }\n",
              "            .column {\n",
              "                flex: 1;\n",
              "                padding: 10px;\n",
              "                white-space: pre-wrap;\n",
              "                text-align: left;\n",
              "            }\n",
              "            .diff-remove {\n",
              "                background-color: #d9534f;  /* Dark red */\n",
              "                color: white;\n",
              "                text-decoration: line-through;\n",
              "                border-radius: 4px;\n",
              "                padding: 2px 4px;\n",
              "            }\n",
              "            .diff-add {\n",
              "                background-color: #5cb85c;  /* Dark green */\n",
              "                color: white;\n",
              "                border-radius: 4px;\n",
              "                padding: 2px 4px;\n",
              "            }\n",
              "        </style>\n",
              "    </head>\n",
              "    <body>\n",
              "        <div class=\"container\">\n",
              "            <div class=\"column\" style=\"border-right: 1px solid #ccc;\">\n",
              "                <br>s = <span class='diff-remove'>\\Big (</span> { \\frac { 4 \\pi } { n } } <span class='diff-remove'>\\Big )</span> \\sqrt { \\gamma ( \\rho - { \\frac { \\Phi \\tilde { \\rho } } { 2 } } - { \\frac { k \\gamma } { a ^ { 2 } } } <span class='diff-remove'>)</span> } .<br><br>\n",
              "            </div>\n",
              "            <div class=\"column\">\n",
              "                <br>s = <span class='diff-add'>\\left(</span> { \\frac { 4 \\pi } { n } } <span class='diff-add'>\\right)</span> \\sqrt { \\gamma ( \\rho - { \\frac { \\Phi \\tilde { \\rho } } { 2 } } - { \\frac { k \\gamma } { a ^ { 2 } } } } <span class='diff-add'>\\right)</span> .<br><br>\n",
              "            </div>\n",
              "        </div>\n",
              "    </body>\n",
              "    </html>\n",
              "    "
            ]
          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<PIL.JpegImagePlugin.JpegImageFile image mode=RGB size=200x50>"
            ],
            "image/png": 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\n",
            "image/jpeg": "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\n"
          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<IPython.core.display.HTML object>"
            ],
            "text/html": [
              "\n",
              "    <html>\n",
              "    <head>\n",
              "        <style>\n",
              "            body { font-family: Arial, sans-serif; margin: 0; padding: 0; }\n",
              "            .container { display: flex; align-items: flex-start; }\n",
              "            .column {\n",
              "                flex: 1;\n",
              "                padding: 10px;\n",
              "                white-space: pre-wrap;\n",
              "                text-align: left;\n",
              "            }\n",
              "            .diff-remove {\n",
              "                background-color: #d9534f;  /* Dark red */\n",
              "                color: white;\n",
              "                text-decoration: line-through;\n",
              "                border-radius: 4px;\n",
              "                padding: 2px 4px;\n",
              "            }\n",
              "            .diff-add {\n",
              "                background-color: #5cb85c;  /* Dark green */\n",
              "                color: white;\n",
              "                border-radius: 4px;\n",
              "                padding: 2px 4px;\n",
              "            }\n",
              "        </style>\n",
              "    </head>\n",
              "    <body>\n",
              "        <div class=\"container\">\n",
              "            <div class=\"column\" style=\"border-right: 1px solid #ccc;\">\n",
              "                <br>f ( t , z , \\bar { z } ) = F _ { <span class='diff-remove'>\\star</span> } \\left( t , \\frac { z } { \\sqrt { 2 \\theta } } , \\frac { \\bar { z } } { \\sqrt { 2 \\theta } } \\right) ,<br><br>\n",
              "            </div>\n",
              "            <div class=\"column\">\n",
              "                <br>f ( t , z , \\bar { z } ) = F _ { <span class='diff-add'>*</span> } \\left( t , \\frac { z } { \\sqrt { 2 \\theta } } , \\frac { \\bar { z } } { \\sqrt { 2 \\theta } } \\right) ,<br><br>\n",
              "            </div>\n",
              "        </div>\n",
              "    </body>\n",
              "    </html>\n",
              "    "
            ]
          },
          "metadata": {}
        }
      ],
      "source": [
        "# @title Suffix vs. generated text\n",
        "\n",
        "for i in range (10):\n",
        "    image, label = test_dataset[i]\n",
        "    prefix = \"<image>\" + label[\"prefix\"]\n",
        "    suffix = label[\"suffix\"]\n",
        "\n",
        "    inputs = processor(\n",
        "        text=prefix,\n",
        "        images=image,\n",
        "        return_tensors=\"pt\"\n",
        "    ).to(TORCH_DTYPE).to(DEVICE)\n",
        "\n",
        "    prefix_length = inputs[\"input_ids\"].shape[-1]\n",
        "\n",
        "    with torch.inference_mode():\n",
        "        generation = model.generate(**inputs, max_new_tokens=256, do_sample=False, num_beams=3)\n",
        "        generation = generation[0][prefix_length:]\n",
        "        generated_text = processor.decode(generation, skip_special_tokens=True)\n",
        "\n",
        "    html_diff = side_by_side_diff_divs(suffix, generated_text)\n",
        "    display(image)\n",
        "    display(HTML(html_diff))"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "5cU-C88_IfAQ"
      },
      "source": [
        "### Evaluate fine-tuned PaliGemma2 model"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 21,
      "metadata": {
        "id": "6zUxJufnIVMK"
      },
      "outputs": [],
      "source": [
        "import numpy as np\n",
        "\n",
        "targets = []\n",
        "predictions = []\n",
        "\n",
        "with torch.inference_mode():\n",
        "    for i in range(10):\n",
        "        image, label = test_dataset[i]\n",
        "        prefix = \"<image>\" + label[\"prefix\"]\n",
        "        suffix = label[\"suffix\"]\n",
        "\n",
        "        inputs = processor(\n",
        "            text=prefix,\n",
        "            images=image,\n",
        "            return_tensors=\"pt\"\n",
        "        ).to(TORCH_DTYPE).to(DEVICE)\n",
        "\n",
        "        prefix_length = inputs[\"input_ids\"].shape[-1]\n",
        "\n",
        "        generation = model.generate(**inputs, max_new_tokens=256, do_sample=False, num_beams=3)\n",
        "        generation = generation[0][prefix_length:]\n",
        "        generated_text = processor.decode(generation, skip_special_tokens=True)\n",
        "\n",
        "        targets.append(suffix)\n",
        "        predictions.append(generated_text)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ulb7SNacJua5"
      },
      "source": [
        "**NOTE:** BLEU (Bilingual Evaluation Understudy) is a metric used to measure the similarity between a VLM's generated text (like a caption for an image) and a human-written reference text. It works by calculating the overlap of words and phrases, giving a score between 0 and 1.  A higher BLEU score indicates better agreement between the generated text and the reference, meaning the VLM is doing a better job of producing expected output."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 22,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "3c2S5avCKiDM",
        "outputId": "d15c19b8-7d85-4c45-a1a2-bf491816caa2"
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "\u001b[?25l   \u001b[90m━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━\u001b[0m \u001b[32m0.0/84.0 kB\u001b[0m \u001b[31m?\u001b[0m eta \u001b[36m-:--:--\u001b[0m\r\u001b[2K   \u001b[90m━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━\u001b[0m \u001b[32m84.0/84.0 kB\u001b[0m \u001b[31m7.5 MB/s\u001b[0m eta \u001b[36m0:00:00\u001b[0m\n",
            "\u001b[?25h\u001b[?25l   \u001b[90m━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━\u001b[0m \u001b[32m0.0/480.6 kB\u001b[0m \u001b[31m?\u001b[0m eta \u001b[36m-:--:--\u001b[0m\r\u001b[2K   \u001b[91m━━━━━━━━━━━━━━━━\u001b[0m\u001b[90m╺\u001b[0m\u001b[90m━━━━━━━━━━━━━━━━━━━━━━━\u001b[0m \u001b[32m194.6/480.6 kB\u001b[0m \u001b[31m10.6 MB/s\u001b[0m eta \u001b[36m0:00:01\u001b[0m\r\u001b[2K   \u001b[91m━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━\u001b[0m\u001b[91m╸\u001b[0m\u001b[90m━━━\u001b[0m \u001b[32m440.3/480.6 kB\u001b[0m \u001b[31m6.5 MB/s\u001b[0m eta \u001b[36m0:00:01\u001b[0m\r\u001b[2K   \u001b[90m━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━\u001b[0m \u001b[32m480.6/480.6 kB\u001b[0m \u001b[31m5.5 MB/s\u001b[0m eta \u001b[36m0:00:00\u001b[0m\n",
            "\u001b[?25h\u001b[?25l   \u001b[90m━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━\u001b[0m \u001b[32m0.0/116.3 kB\u001b[0m \u001b[31m?\u001b[0m eta \u001b[36m-:--:--\u001b[0m\r\u001b[2K   \u001b[90m━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━\u001b[0m \u001b[32m116.3/116.3 kB\u001b[0m \u001b[31m9.9 MB/s\u001b[0m eta \u001b[36m0:00:00\u001b[0m\n",
            "\u001b[?25h\u001b[?25l   \u001b[90m━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━\u001b[0m \u001b[32m0.0/179.3 kB\u001b[0m \u001b[31m?\u001b[0m eta \u001b[36m-:--:--\u001b[0m\r\u001b[2K   \u001b[90m━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━\u001b[0m \u001b[32m179.3/179.3 kB\u001b[0m \u001b[31m15.8 MB/s\u001b[0m eta \u001b[36m0:00:00\u001b[0m\n",
            "\u001b[?25h\u001b[?25l   \u001b[90m━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━\u001b[0m \u001b[32m0.0/134.8 kB\u001b[0m \u001b[31m?\u001b[0m eta \u001b[36m-:--:--\u001b[0m\r\u001b[2K   \u001b[90m━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━\u001b[0m \u001b[32m134.8/134.8 kB\u001b[0m \u001b[31m12.9 MB/s\u001b[0m eta \u001b[36m0:00:00\u001b[0m\n",
            "\u001b[?25h\u001b[?25l   \u001b[90m━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━\u001b[0m \u001b[32m0.0/194.1 kB\u001b[0m \u001b[31m?\u001b[0m eta \u001b[36m-:--:--\u001b[0m\r\u001b[2K   \u001b[90m━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━\u001b[0m \u001b[32m194.1/194.1 kB\u001b[0m \u001b[31m17.0 MB/s\u001b[0m eta \u001b[36m0:00:00\u001b[0m\n",
            "\u001b[?25h\u001b[31mERROR: pip's dependency resolver does not currently take into account all the packages that are installed. This behaviour is the source of the following dependency conflicts.\n",
            "gcsfs 2024.10.0 requires fsspec==2024.10.0, but you have fsspec 2024.9.0 which is incompatible.\u001b[0m\u001b[31m\n",
            "\u001b[0m"
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      "source": [
        "!pip install -q evaluate"
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    },
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        },
        "id": "ATtWcyjFH9EC",
        "outputId": "1fe5cc3d-46ef-4ac3-cf38-2da28a273575"
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      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "Downloading builder script:   0%|          | 0.00/5.94k [00:00<?, ?B/s]"
            ],
            "application/vnd.jupyter.widget-view+json": {
              "version_major": 2,
              "version_minor": 0,
              "model_id": "65de82823814445c90eccec2b8202db1"
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          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "Downloading extra modules:   0%|          | 0.00/1.55k [00:00<?, ?B/s]"
            ],
            "application/vnd.jupyter.widget-view+json": {
              "version_major": 2,
              "version_minor": 0,
              "model_id": "88810a621c9d41449d7dbd0234f22904"
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          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "Downloading extra modules:   0%|          | 0.00/3.34k [00:00<?, ?B/s]"
            ],
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              "version_major": 2,
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          "metadata": {}
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          "text": [
            "{'bleu': 0.7391661684279524, 'precisions': [0.9650455927051672, 0.8796296296296297, 0.7915360501567398, 0.7181528662420382], 'brevity_penalty': 0.886866522219942, 'length_ratio': 0.89280868385346, 'translation_length': 658, 'reference_length': 737}\n"
          ]
        }
      ],
      "source": [
        "# @title Calculate BLEU\n",
        "\n",
        "from evaluate import load\n",
        "\n",
        "bleu = load(\"bleu\")\n",
        "\n",
        "results = bleu.compute(predictions=predictions, references=targets)\n",
        "print(results)"
      ]
    },
    {
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      "execution_count": 24,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "LdWWIf4kKWqs",
        "outputId": "8140f93e-11c1-481b-8a2f-1875260137d6"
      },
      "outputs": [
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          "output_type": "stream",
          "name": "stdout",
          "text": [
            "\u001b[2K     \u001b[90m━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━\u001b[0m \u001b[32m51.8/51.8 kB\u001b[0m \u001b[31m4.7 MB/s\u001b[0m eta \u001b[36m0:00:00\u001b[0m\n",
            "\u001b[2K   \u001b[90m━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━\u001b[0m \u001b[32m104.0/104.0 kB\u001b[0m \u001b[31m9.1 MB/s\u001b[0m eta \u001b[36m0:00:00\u001b[0m\n",
            "\u001b[?25h"
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        }
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      "source": [
        "!pip install -q sacrebleu"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "07DPD2UPKyBR"
      },
      "source": [
        "**NOTE:** Translation Error Rate (TER) is another metric used to evaluate machine translation, often applied to VLMs that generate text. Unlike BLEU, which focuses on overlap, TER calculates the number of edits (insertions, deletions, substitutions, shifts) needed to change the generated text into a human-written reference."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 25,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 66,
          "referenced_widgets": [
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      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "Downloading builder script:   0%|          | 0.00/9.99k [00:00<?, ?B/s]"
            ],
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          "metadata": {}
        },
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "{'score': 17.472698907956318, 'num_edits': 112, 'ref_length': 641.0}\n"
          ]
        }
      ],
      "source": [
        "# @title Calculate TER\n",
        "\n",
        "from evaluate import load\n",
        "\n",
        "ter = load(\"ter\")\n",
        "results = ter.compute(predictions=predictions, references=targets, case_sensitive=True)\n",
        "print(results)"
      ]
    }
  ],
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